Chapter 8 Lecture:
Present Value Mathematics for
Real Estate
Real estate deals almost
always involve cash amounts
at different points in time.
Examples:
– Buy a property now, sell it later.
– Sign a lease now, pay rents monthly over time.
– Take out a mortgage now, pay it back over time.
– Buy land now for development, pay for construction
and sell the building later.
Present Value Mathematics

In order to be successful in a real estate career (not
to mention to succeed in subsequent real estate
courses), you need to know how to relate cash
amounts across time.
 In other words, you need to know how to do
“Present Value Mathematics”,
– On a business calculator, and
– On a computer (e.g., using a spreadsheet like
“Excel”®).
Why is $1 today not
equivalent to $1 a year from
now?…
Present Value Mathematics

Dollars at different points in time are related
by the “opportunity cost of capital” (OCC),
expressed as a rate of return.
 We will typically label this rate, “r”.
Example
If r = 10%, then $1 a year from now is worth:
$1
$1

 $0.909
1  0.10 1.1
today.
Two major types of PV math
problems:


Single-sum problems
Multi-period cash flow problems
8.1 Single-sum formulas…
8.1.1 Single-period discounting & growing
FV = (1+ r) PV
FV
PV =
1+ r
Single-period Discounting
Your tenant owes $10,000 in rent. He wants to
postpone payment for a year. You are
willing, but only for a 15% return. How
much will your tenant have to pay?
FV = ( 1.15 ) $10,000  $11,500
This is the basic building block.
Compound single-period
discounting or growth over
multiple periods…
15% for two periods:
FV = (1.15)(1.15) $10,000  (1.15) 2 $10,000  $13,225
$13,225
PV =
 $10,000
2
(1.15)
8.1.2 Single-sums over
multiple periods.
PV 
FV
1  r 
N
FV = (1+ r )N PV
Single-sums over multiple
periods
You’re interested in a property you think is
worth $1,000,000. You think you can sell
this property in five years for that same
amount. Suppose you’re wrong, and you
can only really expect to sell it for $900,000
at that time. How much would this reduce
what you’re willing to pay for the property
today, if 15% is the required return?
Single-sums over multiple
periods
$1,000,000
 $497,177,
5
1.15
$900,000
 $447,459
5
1.15
$497,177 - $447,459 = $49,718, down to
$950,282.
N
I/YR
PV
PMT
FV
5
15
CPT
0
1000000
Solving for the return or the
time it takes to grow a value…
You can buy a piece of land for $1,000,000.
You think you will be able to sell it to a
developer in about 5 years for twice that
amount. You think an investment with this
much risk requires an expected return of
20% per year. Should you buy the land?…
Solving for the return or the
time it takes to grow a value
1/N
 FV 
r=

 PV 
1/ 5
 $2,000,000 
-1 

 $1,000,000 
- 1  14.87%
No.
N
I/YR
PV
PMT
FV
5
CPT
-1000000
0
2000000
Solving for the return or the
time it takes to grow a value
Your investment policy is to try to buy vacant
land when you think its ultimate value to a
developer will be about twice what you
have to pay for the land. You want to get a
20% return on average for this type of
investment. You need to wait to purchase
such land parcels until you are within how
many years of the time the land will be
“ripe” for development?…
Solving for the return or the
time it takes to grow a value
LN(FV) - LN(PV) LN( 2 ) - LN( 1 )
N =

 3.8
LN(1 + r)
LN( 1.20 )
3.8 years.
N
I/YR
PV
PMT
FV
CPT
20
-1
0
2
Simple & Compound
Interest…
15% interest for two years,
compounded:
For every $1 you start with, you end up with:
(1.15)(1.15) = (1.15)2 = $1.3225.
The “15%” is called “compound interest”.
(In this case, the compounding interval is
annual).
Simple & Compound
Interest…
Note: you ended up with 32.25% more than
you started with.
32.25% / 2 yrs = 16.125%.
So the same result could be expressed as:
– 16.125% “simple annual interest” (no
compounding).
– or 32.25% “simple interest” (for two years).
Simple & Compound
Interest…

Suppose you get 1% simple interest each month.
This is referred to as a “12% nominal annual
rate”, or “equivalent nominal annual rate”
(ENAR). We will use the label “i ” (or “NOM”) to
refer to the ENAR:
 Nominal Annual Rate = (Simple Rate Per
Period)(Periods/Yr)
– i
= (r)(m)
– 12% = (1%)(12 mo/yr)
Simple & Compound
Interest…
Suppose the 1% simple monthly interest is
compounded at the end of every month. Then in 1
year (12 months) this 12% nominal annual rate
gives you:
– (1.01)12 = $1.126825
For every $1 you started out with at the beginning of
the year.
12.00% nominal rate = 12.6825% effective annual
rate
“Effective Annual Rate” (EAR) is aka EAY
(“equiv.ann. yield”).
The relationship between
ENAR and EAR:
m
EAR = (1 + i/m ) - 1
1 /m
i = m [(1+ EAR )
- 1]  ENAR
Rates are usually quoted in ENAR terms.

Example:
What is the EAR of a 12% mortgage?
EAR = (1 + 0.12 / 12 )12 - 1  ( 1.01 )12 - 1  12.6825%

You don’t have to memorize the formulas if you
know how to use a business calculator:
HP-10B
TI-BAII PLUS
CLEAR ALL
I Conv
12 P/YR
NOM = 12 ENTER  
12 I/YR
C/Y = 12 ENTER 
EFF% gives 12.68
CPT EFF = 12.68
Effective Rate = “EFF” = “EAR”
 Nominal Rate = “NOM” = ENAR


“Bond-Equivalent” &
“Mortgage-Equivalent”
Rates…
Traditionally, bonds pay interest semiannually (twice per year).
 Bond interest rates (and yields) are quoted
in nominal annual terms (ENAR) assuming
semi-annual compounding (m = 2).
 This is often called “bond-equivalent yield”
(BEY), or “coupon-equivalent yield”
(CEY). Thus:
2
EAR = (1 + BEY/ 2 ) - 1
What is the EAR of an 8%
bond?

“Bond-Equivalent” &
“Mortgage-Equivalent”
Rates
Traditionally, mortgages pay interest
monthly.
 Mortgage interest rates (and yields) are
quoted in nominal annual terms (ENAR)
assuming monthly compounding (m = 12).
 This is often called “mortgage-equivalent
yield” (MEY) Thus:
12
EAR = (1 + MEY/ 12 ) - 1
What is the EAR of an 8%
mortgage?
Example:
Yields in the bond market are currently 8%
(CEY). What interest rate must you charge
on a mortgage (MEY) if you want to sell it
at par value in the bond market?
Answer:
7.8698%.
EAR = (1 + BEY/ 2 )2 - 1  ( 1.04 )2 - 1  0.0816
MEY  12 [(1 + EAR )1 / 12 - 1]  12 [(1.0816 )1 / 12 - 1]  0.078698
HP-10B
TI-BAII PLUS
CLEAR ALL
I Conv
2 P/YR
NOM = 8 ENTER  
8 I/YR
C/Y = 2 ENTER 
EFF% gives 8.16
CPT EFF = 8.16 
12 P/YR
C/Y = 12 ENTER 
NOM% gives 7.8698
CPT NOM = 7.8698
Example:
You have just issued a mortgage with a 10%
contract interest rate (MEY). How high can
yields be in the bond market (BEY) such
that you can still sell this mortgage at par
value in the bond market?
Answer:
10.21%.
EAR = (1 + MEY/ 12 )12 - 1  ( 1.00833 )12 - 1  0.1047
BEY  2 [(1+ EAR ) - 1]  2 [(1.1047 )
1/ 2
1/ 2
HP-10B
TI-BAII PLUS
CLEAR ALL
I Conv
12 P/YR
NOM = 10 ENTER  
10 I/YR
C/Y = 12 ENTER 
EFF% gives 10.47
CPT EFF = 10.47 
2 P/YR
C/Y = 2 ENTER 
NOM% gives 10.21
CPT NOM = 10.21
- 1]  0.1021
8.2 Multi-period problems…

Real estate typically lasts many years.
– And it pays cash each year.

Mortgages last many years and have monthly
payments.
 Leases can last many years, with monthly
payments.
 We need to know how to do PV math with multiperiod cash flows.
 In general,
The multi-period problem is just the sum of a
bunch of individual single-sum problems:
Example:
PV of $10,000 in 2 years @ 10%, and
$12,000 in 3 years at 11% is:
PV 
$10,000
$12,000

 $8,264.46  $8,774.30  $17,038.76
1  0.102 1  0.113
“Special Cases” of the
Multi-Period PV Problem…




Level Annuity
Growth Annuity
Level Perpetuity
Constant-Growth Perpetuity
“Special Cases” of the
Multi-Period PV Problem
These “special cases” of the multi-period PV
problem are particularly interesting to consider.
 They equate to (or approximate to) many practical
situations:

Leases

Mortgages

Apartment properties
 And they have simple multi-period PV formulas.
 This enables general relationships to be observed
in the PV formulas.

Example:
The relationship between the total return, the
current cash yield, and the cash flow growth
rate:
 “Cap Rate”  Return Rate - Growth
Rate.
8.2.2 The Level Annuity in
Arrears
The PV of a regular series of cash flows, all of the
same amount, occurring at the end of each period,
the first cash flow to occur at the end of the first
period (one period from the present).
PV =
CF 1
CF 2
CF N
+
+
.
.
.
+
(1 + r)
(1 + r )2
(1 + r )N
where CF1 = CF2 = . . . = CFN.
Label the constant periodic cash flow amount:
“PMT ”:
PMT
PMT
PMT
PV =
+
+ ... +
2
(1 + r)
(1 + r )
(1 + r )N
Example:
What is the present value of a 20-year
mortgage that has monthly payments of
$1000 each, and an interest rate of
12%/year?
PV 
$1000 $1000 $1000
$1000





 $90,819
2
3
240
1.01
1.01
1.01
1.01
(Note: 12% nominal rate is 1% per month
simple interest.)
Answer:
$90,819.
This can be found either by applying the
“Annuity Formula”
PV =
PMT
PMT
PMT
+
+
.
.
.
+
(1 + r)
(1 + r )2
(1 + r )N
1 - 1/(1 + r )N
 PMT
r
1 - 1/( 1.01 )240
 $1000
 $90,819
0.01
Answer (cont’d):
Or by using your calculator (“mortgage math
keys”):
N
I/YR
PV
PMT
FV
240
12
CPT
1000
0
Answer (cont’d):
In a computer spreadsheet like Excel®, there are
functions you can use to solve for these variables.
The Excel® functions equivalent to the HP-10B
calculator registers are:
N
I/YR
PV
PMT
FV
=NPER()
=RATE()
=PV()
=PMT()
=FV()
However, the computer does not convert
automatically from nominal annual terms to
simple periodic terms. For example, a 30-year,
12% interest rate mortgage with monthly
payments in Excel® is a 360-month, 1% interest
mortgage.
For example:
The Excel formula below:
=PMT(.01,240,-90819)
will return the value $1000.00 in the cell in which
the formula is entered.
You can use the Excel f x key on the toolbar to get
help with financial functions.
8.2.9 Solving the Annuity for
Future Value.
Suppose we want to know how much a level
stream of cash flows (in arrears) will grow
to, at a constant interest rate, over a
specified number of periods.
Example:
What is the future value of $1000 deposited at
the end of every month, at 12% annual
interest compounded monthly?
(1.01) 239 $1000  (1.01) 238 $1000    $1000  $989,255
N
I/YR
PV
PMT
FV
240
12
0
1000
CPT
The general formula is:
 PMT
PMT
PMT 
FV  (1  r ) PV  (1  r ) 
+
+ ... +
2
N
(1
+
r)
(1
+
r
(1
+
r
)
)


N
N
N


1

1
/(
1

r
)
 (1  r ) N  PMT

r


8.2.10 Solving for the
Annuity Periodic Payment
Amount.
What is the regular periodic payment amount
(in arrears) that will provide a given present
value, discounted at a stated interest rate, if
the payment is received for a stated number
of periods?
Just invert the annuity formula to solve for
the PMT amount:
r
PMT = PV
1 - 1/(1 + r )N
Example:
A borrower wants a 30-year, monthlypayment, fixed-interest mortgage for
$80,000. As a lender, you want to earn a
return of 1 percent per month, compounded
monthly. Then you would agree to provide
the $80,000 up front (i.e., at the present
time), in return for the commitment by the
borrower to make 360 equal monthly
payments in the amount of?…
Answer:
0.01
PMT = $80,000
 $822.89
360
1  1 / 1.01
$822.89
Or, on the calculator (with P/Yr = 12):
N
I/YR
PV
PMT
FV
360
12
80000
CPT
0
8.2.11 Solving for the
Number of Periodic
Payments
How many regular periodic payments (in
arrears) will it take to provide a given return
to a given present value amount?
The general formula is:
PV 

LN  1 - r

PMT


N = LN 1+ r 
where LN is the natural logarithm.
Example:
How long will it take you to pay off a $50,000
loan at 10% annual interest (compounded
monthly), if you can only afford to pay
$500 per month?
Answer:
50000 

LN  1 - (0.10 / 12)

500 

216 = LN 1+ (0.10 / 12) 
Or solve it on the calculator:
N
I/YR
PV
PMT
FV
CPT
10
50000
-500
0
Another example:
Solving for the number of periods required to
obtain a future value.
The general formula is:
FV 

LN  1  r

PMT 

N =
LN 1+ r 
Example:
You expect to have to make a capital
improvement expenditure of $100,000 on a
property in five years. How many months
before that time must you begin setting
aside cash to have available for that
expenditure, if you can only set aside $2000
per month, at an interest rate of 6%, if your
deposits are at the ends of each month?
Answer:
100000 

LN  1  (0.005)

2000 

45 =
LN 1.005
45 months
Or solve it on the calculator:
N
I/YR
PV
PMT
FV
CPT
6
0
-2000
100000
8.2.3 The Level Annuity in
Advance
The PV of a regular series of cash flows, all of
the same amount, occurring at the
beginning of each period, the first cash
flow to occur at the present time.
The Level Annuity in
Advance
This just equals (1+r) times the PV of the
annuity in arrears:
PV = PMT 
PMT
PMT
+ ... +
(1 + r)
(1 + r )N 1
 PMT
PMT
PMT 

 (1  r )
+
+
.
.
.
+
2
N 
(1 + r )
(1 + r ) 
 (1 + r)
1 - 1/(1 + r )N
 (1  r ) PMT
r
1 - 1/(1 + r )N
 PMT (1  r )
r
Example:
What is the present value of a 20-year lease
that has monthly rental payments of $1000
each, due at the beginning of each month,
where 12%/year is the appropriate cost of
capital (OCC) for discounting the rental
payments back to present value?
Answer:
$1000 $1000
$1000
PV  $1000 


1.01
1.012
1.01239

1 - 1/( 1.01 )240 

 $1000 (1.01)

0.01


 $91,728
(Note: This is slightly more than the $90,819
PV of the annuity in arrears.)
(Remember: r = i/m, e.g.: 1% = 12%/yr. 
12mo/yr.)
Answer (cont’d):
In your calculator, set:
HP-10B: Set BEG/END key to BEGIN
TI-BA II+: Set 2nd BGN, 2nd SET, ENTER
(Note: This setting will remain until you
change it back.)
Then, as with the level annuity in advance:
N
I/YR
PV
PMT
FV
240
12
CPT
1000
0
8.2.4 The Growth Annuity in
Arrears
The PV of a finite series of cash flows, each
one a constant multiple of the preceding
cash flow, all occurring at the ends of the
periods.
Each cash flow is the same multiple of the
previous cash flow.
Example:
$100.00, $105.00, $110.25,
$115.76,. . .
$105 = (1.05)$100, $110.15 = (1.05)$105,
etc…
Continuing for a finite number of periods.
Example:
A 10-year lease with annual rental payments
to be made at the end of each year, with the
rent increasing by 2% each year. If the first
year rent is $20/SF and the OCC is 10%,
what is the PV of the lease?
Answer:

$20 1.02 $20 1.02  $20
1.02 $20
PV 



1.10
1.10 2
1.103
1.1010
2
$132.51/SF.
9
How do we know?…
The general formula is:
1  g CF1  1  g  CF1    1  g  CF1
CF1
PV 

(1  r )
(1  r ) 2
(1  r )3
(1  r ) N
2
 1  1  g  1  r N
 CF1 
rg

N 1




So in the specific case of our example:
 1  1.02 / 1.1010 
  $20(6.6253)  $132.51.
PV  $20

0
.
10

0
.
02


Answer (cont’d):
Unfortunately, the typical business calculator
is not set upto do this type of problem
“automatically”. You either have to
memorize the formula (or steps on your
calculator), as below:
1.02 1.1 = 0.9273

yx 10 = 0.47
- 1 = -0.53 +/-
.08 = 6.6253 x 20 = 132.51
Answer (cont’d):
Or you can “trick” the calculator into solving this
problem using its mortgage math keys, by
transforming the level annuity problem:
1. Redefine the interest rate to be: (1+r)/(1+g)-1
(e.g., 1.1/1.02 - 1 = 0.078431 = 7.8431%.
2. Solve the level annuity in advance problem with
this “fake” interest rate (e.g., BEG/END, N=10,
I/YR=7.8431, PMT=20, FV=0, CPT
PV=145.7568).
3. Then divide the answer by 1+r (e.g., 145.7568 /
1.1 = 132.51).
(Note: you set the calculator for CFs in advance
(BEG or BGN), but that’s just a “trick”. The
actual problem you are solving is for CFs in
arrears.)
A more complicated
example (like Study
Qu.#49):
A landlord has offered a tenant a 10-year lease
with annual net rental payments of $30/SF
in arrears. The appropriate discount rate is
8%. The tenant has asked the landlord to
come back with another proposal, with a
lower initial rent in return for annual stepups of 3% per year. What should the
landlord’s proposed starting rent be?
Answer
1)
First solve the level-annuity PV problem:
 1  1 / 1.0810 
  $201.30
PV  $30

0.08


2)
Then plug this answer into the growthannuity formula and invert to solve for
CF1:
 0.08  0.03 
  $26.66
CF1  $201.30
10 
 1  1.03 / 1.08 
Or, using the “calculator trick”
method, the steps are:
HP-10B
TI-BA II+
CLEAR ALL, BEG/END=END
BGN SET (=END) ENTER QUIT
1 P/YR
P/Y = 1 ENTER QUIT
10 N
10 N
8 I/Y
8 I/Y
30 PMT
30 PMT
PV gives 201.30
CPT PV = 201.30
1.08/1.03-1=.0485X100=
1.08/1.03-1=.0485 X 100=
4.85437 I/Y
4.85437 I/Y
BEG/END = BEG
BGN SET (=BGN) ENTER QUIT
PMT gives 24.68713
CPT PMT = 24.68713
X 1.08 = 26.66
X 1.08 = 26.66
Answer (cont’d):
Tenant gets an initial rent of $26.66/SF
instead of $30.00/SF, but landlord gets same
PV from lease (because rents grow).
8.2.5 The Constant-Growth
Perpetuity (in arrears).
The PV of an infinite series of cash flows,
each one a constant multiple of the
preceding cash flow, all occurring at the
ends of the periods.
Like the Growth annuity only it’s a
perpetuity instead of an annuity: an infinite
stream of cash flows.
The general formula is:

CF1
1  g CF1 1  g  CF1
PV 



2
3
(1  r )
(1  r )
(1  r )
2
CF1

rg
The entire infinite sum collapses to this simple
ratio!
This is a very useful formula,
because:
1)
2)
3)
It’s simple.
It approximates a commercial building.
Especially if the building doesn’t have
long-term leases, and operates in a rental
market that is not too cyclical (like most
apartment buildings).
The perpetuity formula &
the cap rate…
You’ve already seen the perpetuity formula.
It’s the cap rate formula:
BldgVal
 PV 
CF1
NOI

r  g caprate

caprate 

r
f
NOI
PV

rg
 RP   g
 Tbill Rate  Risk Pr emium  GrowthRate
(Remember the three determinants of the cap rate…)
Example:
If the the required expected total return on the
investment in the property (the OCC) is 10%,
And the average annual growth rate in the rents and
property value is 2%,
Then the cap rate is approximately 8% (= 10% 2%).
(Or vice versa: If the cap rate is 8% and the average
growth rate in rents and property values is 2%,
then the expected total return on the investment in
the property is 10%.)
Example:
An apartment building has 100 identical units that
rent at $1000/month with building operating
expenses paid by the landlord equal to $500/mo.
On average, there is 5% vacancy. You expect both
rents and operating expenses to grow at a rate of
3% per year (actually: 0.25% per month). The
opportunity cost of capital is 12% per year
(actually: 1% per month). How much is the
property worth?
1) Calculate initial monthly
cash flow:
Potential Rent (PGI) = $1000 * 100 = $100,000
Less Operating Expenses = -500 * 100 = -50,000
---------------- ---------$500 * 100 = $50,000
Less 5% Vacancy0.95 * $50,000 =$47,500 NOI
2) Calculate PV of ConstantGrowth Perpetuity:
$47500 1.0025$47500 1.0025 $47500
PV 



2
3
1.01
1.01
1.01
2
CF1
$47500
$47500



 $6,333,333
r  g 0.01  0.0025 0.0075
8.2.12 Combining the Single
Lump Sum and the Level
Annuity Stream: Classical
Mortgage Mathematics
The typical calculations associated with
mortgage mathematics can be solved as a
combination of the single-sum and the
level-annuity (in arrears) problems we
have previously considered.
The classical mortgage is a
level annuity in arrears with:
Loan Principal = “PV” amount
Regular Loan Payments = “PMT” amount
Outstanding Loan Balance
(OLB) can be found in either
of two (mathematically
equivalent) ways:
OLB = PV with “N” as the
number of remaining (unpaid) regular
payments.
2)
OLB = FV with “N” as the
number of already paid regular payments.
1)
OLB
In other words, if Tm is the original number
of payments in a fully-amortizing loan, and
“q” is the number of payments that have
already been made on the loan, then the
OLB is given by the following general
formula:


(Tm- q)
 1 

1- 
 1+ i 


m


OLB = PMT
i
m
Example:
Recall the $80,000, 30-year, 12% mortgage
with monthly payements that we previously
examined (recall that PMT = $822.89 per
month). What is its OLB after 10 years of
payments?
Answer:
1 - 1/(1.01 )240
74734 = 822.89
.01
$74,734.
This can be computed in
either of two ways on your
calculator:
1)
2)
Solve for PV with N=number of
remaining payments:
N
I/YR
PV
PMT
FV
240
12
CPT
822.89
0
Solve for FV with N=number of payments
already made:
N
I/YR
PV
PMT
FV
120
12
80000
822.89
CPT
Another example: “Balloon
Mortgages”…
Some mortgages are not “fully amortizing”,
that is, they are required to be paid back
before their OLB is fully paid down by the
regular monthly payments.
The “balloon payment” (due at the maturity
of the loan, when the borrower is required
to pay off the loan), simply equals the OLB
on the loan at that point in time.
Example:
What is the balloon payment owed at the end
of an $80,000, 12%, monthly payment
mortgage if the maturity of the loan is 10
years and the amortization rate on the loan
is 30 years?
Answer:
240
1 - 1/(1.01 )
74734 = 822.89
.01
$74,734 (sounds familiar?)…
Found either as:
N
I/YR
PV
PMT
FV
240
12
CPT
822.89
0
N
I/YR
PV
PMT
FV
120
12
80000
822.89
CPT
Or:
Interest-only mortgages:
Some mortgages don’t amortize at all. That is, they
don’t pay down their principal balance at all
during the regular monthly payments. The OLB
remains equal to the initial principal amount
borrowed, throughout the life of the loan. This is
called an “interest-only” loan.
You can represent interest-only loans in your
calculator by setting the FV amount equal to the
PV amount (only with an opposite sign),
representing the contractual principal on the loan.
Example:
What is the monthly payment on an $80,000,
12%, 10-year interest-only mortgage?
Answer:
$800 = $80000 (0.12/12) = $80000 (0.01) =
$800.00.
Or compute it on your calculator as:
N
I/YR
PV
PMT
FV
120
12
80000
CPT
-80000
8.2.12 How the Present
Value Keys on a Business
Calculator Work
There are two types of PV Math keys on a
typical business calculator:
1) The “Mortgage Math”
keys.
They are useful for problems that involve a level
annuity and/or at most one additional single-sum
amount at the end of the annuity (e.g., the typical
mortgage, even if it is not fully-amortizing). These
keys solve the following equation:
 PMT
PMT
PMT 
FV

PV = 
+
+
.
.
.
+
+
2
(1 + r )N  (1 + r )N
 (1 + r) (1+ r )
Normally:
PV = Loan initial principal or OLB.
FV = Loan OLB or balloon.
PMT = Regular monthly payment.
r = i/m = Periodic simple interest (per month).
N = Number of payment periods (remaining, or
already paid, depending on how you’re using the
keys).
Note that this equation is
equivalent to:
 PMT
PMT
PMT 
FV
+
0 = - PV + 
+
+ . . .+
2
N
N
(1
+
r)
(1
+
r
(1
+
r
(1
+
r
)
)
)


This makes it clearer why the “PV” amount is
usually opposite sign to the “PMT” and
“FV” amounts. (If you are solving for FV
then FV may be of opposite sign to PMT,
and same sign as PV.)
The second type of PV Math
keys on your calculator are…
2) The “DCF keys”. They are useful for problems
that are not a simple or single level annuity. The
DCF keys can take an arbitrary stream of future
cash flows, and convert them to present value or
compute the Internal Rate of Return (“IRR”) of
the cash flow stream (including an up-front
negative cash flow representing the initial
investment). The DCF keys solve the following
equation:
NPV = CF 0 +
CF 1
CF 2
CF T
+
+
.
.
.
+
(1 + IRR)
(1 + IRR )2
(1 + IRR )T
The DCF Keys
where the IRR is the discount rate that implies
NPV=0. Typically, CF0 is the up-front
investment and therefore is of opposite sign
to most of the subsequent cash flows (CF1,
CF2,…). Each cash flow can be a different
amount.
The DCF Keys
In general, the IRR is the single rate of return
(per period) that causes all the cash flows
(including the up-front investment) to
discount such that NPV=0.
The DCF Keys
By setting CF0=0, and entering the opportunity cost
of capital (the discount rate, or required “going-in
IRR”) as the interest rate (“I/YR” in the HP-10B,
or “I” in the “NPV” program of the TI-BA2+), the
DCF keys will solve the above equation (with IRR
equal to the input interest rate) for the present
value of the future cash flow stream.
Note: Make sure that the last cash flow amount,
CFT, includes the “reversion” (or asset terminal
value) amount.
Example (like Ch.8 Study
Qu.#53):
Suppose a certain property is expected to produce
net operating cash flows annually as follows, at
the end of each of the next five years: $15,000,
$16,000, $20,000, $22,000, and $17,000. In
addition, at the end of the fifth year the property
we will assume the property will be (or could be)
sold for $200,000. What is the net present value
(NPV) of a deal in which you would pay $180,000
for the property today assuming the required
expected return or discount rate is 11% per year?
Answer:
+$4,394 (the property is worth $184,394.)
NPV = PV(Benefit) – PV(Cost) = $184,394
- $180,000 = +$4,394.
 15000   16000   20000   22000   (17000 + 200000) 
4394 = - 180000 + 
+
+
+
+
2
3
4

(1 + 0.11 )5
 1+ 0.11   (1 + 0.11 )   (1 + 0.11 )   (1 + 0.11 )  

On the calculator this is solved
using the DCF keys as:
HP-10B
TI-BAII+
CLEAR ALL
2nd P/Y = 1 ENTER 2nd QUIT
1 P/YR
CF
11 I/YR
2nd CLR_Work
180000 +/- CFj
CF0 = 180000 +/- ENTER
15000 CFj
C01 = 15000 ENTER 
16000 CFj
F01 = 1 
20000 CFj
C02 = 16000 ENTER 
22000 CFj
F02 = 1 
17000+200000=217000 CFj
C03 = 20000 ENTER 
NPV gives 4394
F03 = 1 
C04 = 22000 ENTER 
F04 = 1 
C05 = 17000+200000=217000 ENTER
F05 = 1
NPV I= 11 ENTER 
NPV CPT = 4394
Example (cont.):
If you could get the previous property for only
$170,000, what would be the expected IRR
of your investment?
Answer:
 
  (17000 + 200000) 
20000  
22000
 15000   16000




+

0 = - 170000 + 
+
+
+
2 
3 
4 
5
1
+
0.1315

  (1 + 0.1315 )   (1 + 0.1315 )   (1 + 0.1315 )   (1 + 0.1315 ) 
13.15%:
HP-10B
TI-BAII+
CLEAR ALL
2nd P/Y = 1 ENTER 2nd QUIT
1 P/YR
CF
170000 +/-
2nd CLR_Work (if you want)
15000 CFj
CF0 = 170000 +/- ENTER
16000 CFj
C01 = 15000 ENTER 
20000 CFj
F01 = 1 
22000 CFj
C02 = 16000 ENTER 
17000+200000=217000 CFj
F02 = 1 
IRR gives 13.15
C03 = 20000 ENTER 
F03 = 1 
C04 = 22000 ENTER 
F04 = 1 
C05 = 17000+200000=217000
ENTER
F05 = 1
IRR CPT = 13.15